My house was completely destroyed by fire. I’m trying to collect on my personal property that I had in the house, but the insurance company is telling me I need an inventory. Can they do that?

Yes. Whether your policy pays for the replacement or just the actual cash value, the company is only obligated to pay for personal property that you can prove you owned at the time of loss. You should keep an up to date inventory or maybe take a video of all your possessions.